Since the discriminant of the quadratic equation Δ is equals to 0, the given quadratic equation has one solution. Thus, the fourth option is correct.
We know that for a quadratic equation [tex]ax^{2} +bx+c =0[/tex], there exists a discriminant, Δ = [tex]b^{2}-4ac[/tex]. This discriminant gives us the nature of roots that a quadratic equation can have.
If Δ > 0, the quadratic equation has two real rootsIf Δ = 0, the quadratic equation has single solution as two double rootsIf Δ < 0, the quadratic equation has no real rootsHere, we have the given quadratic equation as -
[tex]x^{2} -16x+64=0[/tex]
Comparing this to the general expression of a quadratic equation, we have
a = 1
b = -16
c = 64
So, now we have to find the discriminant of the given quadratic equation.
Δ = [tex]b^{2}-4ac[/tex]
[tex]= (-16)^{2} -4*1*64\\= 256-256 = 0[/tex]
We see that the discriminant, Δ = 0. This implies that the given quadratic equation has one solution. Thus, the fourth option is correct.
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A certain element has a half-life of approximately 15 hours. How long would it take for 500 grams of the element to decay to 299 grams? Leave your answer as an integer or simplified expression.
It would take 11.1 hours to decay from 500 grams to 299 grams.
How long takes to decay?We know that for an element with an half-life T has a decay equation that can be written as:
f(t) = A*e^(-t*ln(2)/T)
Where A is the initial amount, and f(t) is the amount of the element t hours after.
Here we know that:
A = 500g
T = 15h
Then the function is:
f(t) = 500g*e^(-t*ln(2)/15h)
And we want to find the value of t such that f(t) = 299g, then:
299g = 500g*e^(-t*ln(2)/15h)
299/500 = e^(-t*ln(2)/15h)
ln(299/500) = -t*ln(2)/15h
-ln(299/500)*15h/ln(2) = t
11.1h = t
It will take 11.1 hours.
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how many different rectangles are there
29. Write the equation of the line that passes
through the point (-6, 1) and has a slope
of ½.
Answer:
Step-by-step explanation:
M is slope
B is y intercept
X and y are given
using the equation for the y intercept
b=y1-mx1
B=1−(1/2)⋅(−6)=4
Write in y=Mx+b
=12+4
The general equation of the line is −2+8=0
Spencer, a professional golfer, had an average score of 80 points last season. This season, however, his average score has been 5% higher. What is Spencer's current average score?
please
Answer:
84
Step-by-step explanation:
80 x 0.05 = 4
80 + 4 =84
a. Find the derivative function f' for the function f.
b. Determine an equation of the line tangent to the graph of f at (a,f(a)) for the given value of a.
The derivative function f' for the function f is f'(x) = -10 / (5x+3)^2 and Equation of the line tangent to the graph of f is 5x+2y+7=0.
Given function:
f(x) = 2/5x+3
a.
f'(x) = d/dx(2/5x+3)
According to power rules:
1/u = -1/u^2
= 2(1/(5x+3)^2) d/dx(5x+3)
= 2*5 / (5x+3)^2
= -10/(5x+3)^2
Hence derivative function f' for the function f is f'(x) = -10 / (5x+3)^2.
b.
(a,f(a)) and a = -1
f(-1) = 2/5(-1)+3
= 2/-5+3
= 2/-2
= -1
point (-1,-1)
slope f'(x) = -10/(5x+3)^2
f'(-1) = -10/(5(-1)+3)^2
= -10/(-2)^2
= -10/4
m = -5/2
Equation of the line is y - y1 = m(x-x1)
y - (-1) = -5/2(x-(-1)
2(y + 1) = -5x - 5
5x + 2y + 2 + 5 = 0
5x+2y+7=0.
Equation of the line tangent to the graph of f is 5x+2y+7=0.
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Add the rational expressions. Show all work.
5/(x+3)(x-4) + 7/(x+2)(x-4)
The given rational expression when added gives
[tex] \frac{12x + 31}{(x + 3)(x - 4)(x + 2)} [/tex]
How to add rational expression?Given:
[tex] \frac{5}{(x + 3)(x - 4)} + \frac{7}{(x + 2)(x - 4)} [/tex]
Find the lowest common multiple (LCM)=
[tex] \frac{5(x + 2) + 7(x + 3)}{(x + 3)(x - 4)(x + 2)} [/tex]
Open parenthesis=
[tex] \frac{5x + 10 + 7x + 21}{(x + 3)(x - 4)(x + 2)} [/tex]
Collect and add like terms=
[tex] \frac{12x + 31}{(x + 3)(x - 4)(x + 2)} [/tex]
So therefore, the addition of the rational expression is
[tex] \frac{12x + 31}{(x + 3)(x - 4)(x + 2)} [/tex]
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Given the equation F=95C+32 where C is the temperature in degrees Celsius and F is the corresponding temperature in degrees Fahrenheit, and the following ordered pairs:
(−25,F1), (−5,F2)
The values of F1 and F2 are 77 and -22 degrees Fahrenheit respectively.
How to use the equation to convert from degrees Celsius to degrees Fahrenheit?Given that; the equation F=9/5C+32 and ordered pairs are (25, F1), (−30, F2)
To convert the values to degrees Fahrenheit, we have to substitute the given values into the equation to get F1 and F2 respectively:
For (25, F1);
F = 9/5 C+32
Now put C= 25 in the equation;
F1 = 9/5(25) + 32
F1 = 45 +22
F1 = 77 degrees Fahrenheit
For (−30, F2);
F = 9/5 C+32
Now put C= -30 in the equation;
F2 = 9/5(-30) + 32
F2 = -54 + 32
F2 = -22 degrees Fahrenheit
Hence, the values of F1 and F2 are 77 and -22 degrees Fahrenheit respectively.
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Which point is located on the line represented by the equation y + 4 = –5(x – 7)?
(-4, 7)
(-7, 4)
(7, -4)
(4, -7)
Answer:
(7,-4)
Step-by-step explanation:
y + 4 = -5(x-7) Substitute in 7 for x and -4 for y
-4 + 4 = -5 (7-7)
0 = 5(0)
0 = 0 This is a true statement.
ne
Problem 14:
From this diagram, select the
pair of lines that must be
parallel if 45 47. If there
is no pair of lines, select
"none."
Playback
2
7
4
3
(first taught in
lesson 24)
9
8
6
After you pick your answer press GO.
A. l || n
B. oll q
C. l | m
D. pllq
E. None
10
7
GO
l
Contro
Lecture
& Problem
Scratchpa
Wallpaper
From the given pair of parallel lines , the lines which satisfies the condition of two angles to be congruent that is ∠5 ≅ ∠7 are l || m .
As given in the question,
Given pair of lines,
Condition given that two angles are congruent ,
∠5 ≅ ∠7
Angle 5 is congruent to vertical opposite angle and to make a pair of parallel line Opposite angle should be congruent to angle 7.
Opposite angle and Angle 7 are pair of corresponding angles.
Line l should be parallel to the line m to satisfied the condition of
∠5 ≅ ∠7.
Therefore, from the given pair of parallel lines , the lines which satisfies the condition of two angles to be congruent that is ∠5 ≅ ∠7 are l || m .
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A group of people are asked how many hours of television they watch each day to the nearest hour. Here are the results:
3 1 2 0 0 1 0 3 2 4
Which of the following is true about this data set?
Answer:
2 people watch tv 3 hours a day, 3 people watch tv 0 hours a day, 2 people watch tv 1 hour a day, 2 people watch tv 2 hours a day, and 1 person watches tv 4 hours a day
Step-by-step explanation:
hope it helped :)
Given the image of the two triangle, which method can you use to show △LHJ≅△KIE
Answer:
ASA
Step-by-step explanation:
As you can see in the picture it is giving us 2 angles, in between those 2 angles are aside. Therefore it is ASA.
We can also eliminate SSS because 2 angles and shown and SAS can also be eliminated because there are 2 angles, not sides.
121x
10
6
4
909
-6-5-4-3-2-12- 2 3 4 5 6 x
J
-8-
-10
B
h
Which statement is true regarding the functions on the
graph?
Of(6) = g(3)
Of(3) = g(3)
Of(3) = g(6)
Of(6) = g(6)
Answer:
The Question is not Correct but answer is f 3 and g 3
On average, a banana will last 6 days from the time it is purchased in the store to the time it is too rotten to eat. Is the mean time to spoil less if the banana is hung from the ceiling? The data show results of an experiment with 15 bananas that are hung from the ceiling. Assume that that distribution of the population is normal.
6.3, 3.5, 5.6, 4.2, 4.4, 3.9, 4.5, 5.5, 7.1, 5.6, 6.6, 6.1, 6.3, 7.2, 4
What can be concluded at the the
α
= 0.01 level of significance level of significance?
For this study, we should use
t-test for a population mean
The null and alternative hypotheses would be:
H
0
:
?
Select an answer
H
1
:
?
Select an answer
The test statistic
?
=
(please show your answer to 3 decimal places.)
The p-value =
(Please show your answer to 4 decimal places.)
The p-value is
?
α
Based on this, we should
Select an answer
the null hypothesis.
Thus, the final conclusion is that ...
The data suggest that the population mean time that it takes for bananas to spoil if they are hung from the ceiling is not significantly less than 6 at
α
= 0.01, so there is statistically insignificant evidence to conclude that the population mean time that it takes for bananas to spoil if they are hung from the ceiling is less than 6.
The data suggest the population mean is not significantly less than 6 at
α
= 0.01, so there is statistically insignificant evidence to conclude that the population mean time that it takes for bananas to spoil if they are hung from the ceiling is equal to 6.
The data suggest the populaton mean is significantly less than 6 at
α
= 0.01, so there is statistically significant evidence to conclude that the population mean time that it takes for bananas to spoil if they are hung from the ceiling is less than 6.
Considering the situation described, it is found that:
A t-test for a population mean should be used.The null hypothesis is: [tex]H_0: \mu = 6[/tex]The alternative hypothesis is: [tex]H_1: \mu < 6[/tex]The test statistic is of: t = -1.94.The p-value is of: 0.0364.The conclusion is of: The data suggest the population mean is not significantly less than 6 at α = 0.01, so there is statistically insignificant evidence to conclude that the population mean time that it takes for bananas to spoil if they are hung from the ceiling is equal to 6.What are the hypothesis tested?At the null hypothesis, it is tested if the mean is of 6 minutes, that is:
[tex]H_0: \mu = 6[/tex]
At the alternative hypothesis, it is tested if the mean is less than 6, hence:
[tex]H_1: \mu < 6[/tex]
What is the test statistic?The test statistic is obtained as follows:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
The parameters of the equation are:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.s is the standard deviation of the sample.n is the sample size.Using a calculator from the sample, the values of the parameters are:
[tex]\overline{x} = 5.39, \mu = 6, s = 1.22, n = 15[/tex]
Hence the test statistic is:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
[tex]t = \frac{5.39 - 6}{\frac{1.22}{\sqrt{15}}}[/tex]
t = -1.94.
What is the p-value and the conclusion?The p-value is obtained with a t-distribution calculator, with a left-tailed test, as we are testing if the mean is less than a value, with t = -1.94 and 15 - 1 = 14 df, hence it is of 0.0364.
Since the p-value is greater than the significance level of 0.01, the null hypothesis is not rejected and there is not enough evidence that the mean is less than 6 days.
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Write the equation of the line that perpendicular to y=5/2x + 9/2 that passes through the point (-5,3)
Answer:
y = -2/5 x + 1
Step-by-step explanation:
slope = -2/5
y-3 = -2/5 (x -(-5)
y-3 = -2/5 (x+5)
y - 3 = -2/5 x - 2
y = -2/5 x - 2 + 3
y = -2/5 x + 1
Please select ONLY ONE of the two questions below to respond to. Choose the one you feel
you can answer the best! This portion of your test will be graded after you submit your exam.
Rubric / Requirements:
1) Make sure your response answers each part of the prompt and is written in at least 3 complete
sentences. (2 points)
2) Proof read your response to ensure you have used proper capitalization and spelling. (1 point)
3) Explain the WHY or HOW behind your ideas! Justify your stances using your knowledge from this unit.
(7 points)
4) Plagiarism in any form is not permitted. You risk a 0% on this section of the exam if you plagiarize.
5) You may use a speech to text tool for your response.
OPTION 1: Political Factions
In his Farewell Address in 1796, George Washington gave the following warnings about the dangers of political
parties:
"[Parties always] distract the public councils and (weaken] the public administration. It agitates the community with ill-
founded jealousies and false alarms, kindles the animosity of one party against another, [encourages) occasionally riot
and insurrection. It opens the door to foreign influence and corruption, which finds access to the government itself
through the channels of party passions..."
QUESTION: Why did George Washington caution against the formation of political parties?
What potential risks do political parties pose towards democracy? Do you think that the
country should have followed his advice?
Answer of the why did George Washington caution against the formation of political parties?
What potential risks do political parties pose towards democracy? Do you think that the country should have followed his advice are as follows :
(a) He believes that the government was undermined by differences between the political parties. Additionally, he argues that "the alternate domination" of one party over another and concurrent attempts to exact revenge on their rivals have resulted in terrible atrocities and "is itself a frightful despotism."
(b)The democratic theory's main tenet is that democracies force administrations to take citizens' preferences into account. Every contemporary democracy is organized by political parties, and some observers contend that parties are what makes democracies responsive. Others, however, contend that parties give extremists a platform and make governments less receptive to the needs of the populace. As newly established democracies around the world grapple with concerns of representation and governability, the discussion of parties and democracy assumes fresh prominence.
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Select the correct answer from each drop-down menu.
The decimal equivalent of the fraction 2/15 is ____. The number of digits that should be repeating are __.
For the first blank options, 0.13, 0.1313, 0.133133, 0.1333
For the second blank options, 3, 2, 1, 0
Answer: 1. 0.1333 2. 1
Step-by-step explanation:
Step 1: 2/15 is the same as 2 divided by 15 which is 1.33333333333333333333333333333333333333333333333333 and so on.
Step 2: Clearly, the only number that is repeating is the 3.
Beth's family is on a road trip. After driving for a while, they stopped for gas, and then traveled 65 1/4 more miles before they decided to stop for lunch. After lunch, they tripled their distance traveled before stopping for the night. Their total distance traveled that day was 372 miles. How many miles had they traveled before their first stop for gas?
Enter your answer as a simplified mixed number in the box.
The distance travelled before their first gas stop is 76.68 miles.
What is a mixed fraction?A fraction consisting of a quotient and remainder is a mixed fraction. we can convert the mixed fraction to improper fraction by first dividing the numerator by denominator and then taking the quotient as whole number and remainder as the numerator of proper fraction keeping the denominator same.
Consider that the distance covered before their first stop for gas be 'x'
The Total distance travelled is 372 miles
WE have given that after stopping for gas they travelled 65 1/4 miles and then tripled their distance. The equation is;
x + 261/4 + 3x = 372
4x + 261/4 = 372
16x + 261 = 1488
16x = 1227
x = 76.68 miles
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The Jellybean jar has a radius of 6.2 cm and a height of 18.3 cm. What would be a
reasonable upper limit for the number of jellybeans in the jar? Reminder: Cylinder's
Volume = pi'r^2*h.
1
100
10000
100000
Please help
The volume or the upper limit of the jellybean jar is 2,208.83cm³.
What do we mean by Volume?The space occupied within an object's borders in three dimensions is referred to as its volume.
It is sometimes referred to as the object's capacity.
Volume and mass are the two fundamental characteristics of matter.
Volume is only the amount of space that a thing occupies.
There are a few methods for determining an object's volume depending on its physical state.
So, the volume of the Jellybean jar:
Formula: πr²h
Now, insert values and calculate as follows:
πr²h
π(6.2)²18.3
π38.44(18.3)
π703.452
2,208.83
Therefore, the volume or the upper limit of the jellybean jar is 2,208.83cm³.
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In a garden club, 90% is ladies. The number of ladies is 12 more than 3 times the number of gentlemen.
How many ladies and how many gentlemen are in the club?
(Note: set up a rational equation and solve)
The number of ladies in the club = 18
and the number of gentlemen in the club = 2
Let x be the number of gentlemen.
So, the number of ladies would be 3x + 12
Total number of people in garden = x + (3x + 12)
= 4x + 12
In a garden club, 90% is ladies.
This means, (3x + 12) is 90 percent of (4x + 12)
We get an equation,
(3x + 12) = 90/100 * (4x + 12)
We solve above equation to find the value of x.
30x + 120 = 36x + 108
6x = 12
x = 2 (number of gentlemen)
So, the number of lades would be:
3x + 12 = 3(2) + 12
= 18
Therefore, the number of ladies in the club = 18 and the number of gentlemen in the club = 2
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Madison wanted to go to the mall, but first she attended her brother Zane's basketball
game. The basketball game lasted 135 minutes, or 15 minutes longer than double the
time Madison spent at the mall. How long was Madison at the mall?
Answer: 60 minutes
Step-by-step explanation:
x=time Madison spent at the mall
2x + 15 = 135
2x = 135-15
x = 120/2 = 60
Answer to each expression
(3 + 9) x 2 dived 4
Answer: [tex]6\\[/tex]
Step-by-step explanation:
[tex]\frac{(3+9) * 2}{4}[/tex]
Add 3 and 9 to get 12.
[tex]\frac{12 * 2}{4}[/tex]
Multiply 12 and 2 to get 24.
[tex]\frac{24}{4}[/tex]
Divide 24 by 4 to get 6.
[tex]6[/tex]
a. {(−5,4), (−4, −1), (−2,1), (0,4), (1,3)
What is the domain and range and it is a function
b. {(−3, −4), (−1,2), (0,0), (−3,5), (2,4)}
Domain:
Range:
Function?
The domain is {-5, -4, -2, 0, 1} and range is {4, -1, 1, 4, 3}
The given relation is {(−5,4), (−4, −1), (−2,1), (0,4), (1,3)
We need to find its domain and range
Domain of the set is {-5, -4, -2, 0, 1}
Range of the set is {4, -1, 1, 4, 3}
If in an relation every input has an single output, it is said to be a function
Here, every element of the domain has a single element in range, so it is a function
The given relation is {(−3, −4), (−1,2), (0,0), (−3,5), (2,4)}
We need to find its domain and range
Domain of the set is {-3, -1, 0, -3, 2}
Range of the set is {-4, 2, 0, 5, 4}
If in an relation every input has an single output, it is said to be a function
Here, every element of the domain has a single element in range, so it is a function
Therefore, the domain is {-5, -4, -2, 0, 1} and range is {4, -1, 1, 4, 3}
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50 percent of 25 percent of the number is 96. What is the number?
Answer: 48
Step-by-step explanation:
first you divide 25/2 and you get 12.5 and that is irrelevant but what is relevant is that you divide 96 by 2 and you get 48 thus giving you the answer to that question. :)
Question 1
Given the demand function D(p) = √/100 - 2p
Find the Elasticity of Demand at a price of $6.
At this price we would say demand is
Inelastic
Unitary
Elastic
Based on this to increase revenue we should
Lower prices
Keep prices unchanged
Raise prices
Answer:
At this price, i would say the demand is: Unitary Inelastic Elastic Based on this, to increase revenue we should: Lower Prices Raise Prices Keep Prices
5. Jen and Kevin and their children: Jon
(age 14), Beth (age 8), and Joshua
(age 4) are visiting a science museum.
Admission prices are shown below.
Museum Admission
Adult (18-64)
Youth (12-17)
Child (5-11)
Under 5
$22.50
$16.25
$9.75
Free
What will be the total cost of admission
for the family?
?
Answer:
$48.5
Step-by-step explanation:
From the question, we know that Jen and Kevin are definitely adults and the fee for adults
= $22.50
Jon is 14, and any number from 12 to 17 is classified as a youth, so his fee is
= $16.25
Beth is 8, classified as a child, her fee is
= $9.25
Joshua's age which is below 5 is granted free entry without a fee
Hence, the total cost of admission for the family is
$22.50 + $16.25 + $9.75
= $48.5
If angle A is (2x) and angle B is (3x+11)° and the sum of angle A and angle B is 66°, find the measure of angle B.
11
22
44
25
The angle B is 66°
Given,
In the question:
If angle A is (2x) and angle B is (3x+11)°
and, the sum of angle A and angle B is 66°
To find the angle of B.
Now, According to the question:
Based on the given condition:
Angle A = 2x
Angle B = (3x + 11)°
The sum of angle A and B is 66°
2x + (3x + 11)° = 66°
5x + 11 = 66°
Calculate the sum or difference
5x = 66 - 11
3x = 55
x = 55/3
The angle B is = 3x + 11
Plug the value of x in angle B
3 × 55/3 + 11
= 66
Hence, The angle B is 66°
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An aircraft (at Z) is spotted by two observers (at X and Y) who are L = 1050 feet apart..
airplane passes over the line joining them, each observer takes a sighting of the angle
elevation to the plane, as indicated in the figure. If A = 40 degrees and B = 30°, how high is th
airplane?
The airplane is 359.12 feet high.
Given,
The distance between the observers = 1050 feet
When an aero plane crosses the line connecting them, each observer measures the plane's elevation angle.
Angle A = 40 degrees
Angle B = 30 degrees
We have to find the height of the airplane.
Here,
C = 180 - (40 + 30) = 110°
According to the sine rule :
Sin110 / 1050 = Sin40 / b = Sin 30 / a
1050 sin40 / sin 110 = b = 718.24 ft
1050 sin 30 / sin 110 = a = 558.69 ft
Sin 40 = h / 558.69
h = 558.69 × sin 40
h = 359.12 feet
That is,
The airplane is 359.12 feet high.
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En la pecera de Simón hay 6 veces la cantidad de peces de colores que peces guppy. Hay un total de 21 peces en la pecera. ¿Cuántos peces de colores más que peces guppy hay?
Answer:god is good
Step-by-step explanation:if u put this 1 stars u hate god
if u give 5 stars u love god
What are the values of x, y, and z?
I need help with this please help me
Answer:
X - 111
Y - 104
Z - 76
Step-by-step explanation:
X - Just add 45 and 66
Y - 180 minus 76
Z - 119 minus 43
NO LINKS! Please help me with this problem #4l
Answer:
equation: (x -1)²/64 -(y -4)²/80 = 1foci: (-11, 4), (13, 4)Step-by-step explanation:
You want the steps to finding the equation of the hyperbola with center (1, 4), vertices (-7, 4) and (9, 4) and that includes the point (-11, -6).
Equation of a hyperbolaThe standard-form equation of a hyperbola with center (h, k) and semi-axes 'a' and 'b' is ...
[tex]\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1[/tex]
The "linear eccentricity" 'c' is the distance from the center to a focus, and satisfies the equation ...
c² = a² +b²
The vertices are (h±a, k) and the foci are (h±c, k).
ApplicationThe center of the hyperbola is given as (1, 4). The distance from the right vertex to the center is ...
a = 9-1 = 8
The equation thus far is ...
(x -1)/8² -(y -4)/b² = 1
The value of 'b' can be determined from the given point:
(-11 -1)²/8² -(-6 -4)²/b² = 1
9/4 -100/b² = 1
5/4 = 100/b²
b² = 100/(5/4) = 80
The linear eccentricity is ...
c² = a² +b²
c² = 64 +80 = 144
c = √144 = 12
The foci are (1±12, 4) = (-11, 4) and (13, 4).
The equation is ...
(x -1)²/64 -(y -4)²/80 = 1
Step summaryThe given information was used to find semi-major axis 'a'. Together with the given center value (h, k), and the given point, the equation was written and solved for b².The value of 'c' was found from a² and b², and used to find the locations of the foci.
Answer:
[tex]\dfrac{(x-1)^2}{64}-\dfrac{(y-4)^2}{80}=1[/tex]
Center = (1, 4)Vertices = (-7, 4) and (9, 4)Foci = (-11, 4) and (13, 4)Step-by-step explanation:
Standard equation of a horizontal hyperbola (opening left and right):
[tex]\boxed{\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1}[/tex]
where:
Center = (h, k)Vertices = (h±a, k)Co-vertices = (h, k±b)Foci = (h±c, k) where c² = a² + b²Given:
Center = (1, 4)Vertices = (-7, 4) and (9, 4)Point on the hyperbola = (-11, -6)Therefore:
h = 1k = 4Find the value of a using the x-values of the vertices:
[tex]\begin{aligned}\implies h-a &= -7\\1-a &= -7\\a &= 8\end{aligned}[/tex]
[tex]\begin{aligned}\implies h+a &= 9\\1+a &= 9\\a &= 8\end{aligned}[/tex]
Therefore, a = 8.
Substitute the values of h, k and a into the formula:
[tex]\implies \dfrac{(x-1)^2}{8^2}-\dfrac{(y-4)^2}{b^2}=1[/tex]
[tex]\implies \dfrac{(x-1)^2}{64}-\dfrac{(y-4)^2}{b^2}=1[/tex]
Substitute the given point on the hyperbola (-11, -6) into the equation and solve for b²:
[tex]\implies \dfrac{(-11-1)^2}{64}-\dfrac{(-6-4)^2}{b^2}=1[/tex]
[tex]\implies \dfrac{(-12)^2}{64}-\dfrac{(-10)^2}{b^2}=1[/tex]
[tex]\implies \dfrac{144}{64}-\dfrac{100}{b^2}=1[/tex]
[tex]\implies \dfrac{144}{64}-1=\dfrac{100}{b^2}[/tex]
[tex]\implies \dfrac{144}{64}-\dfrac{64}{64}=\dfrac{100}{b^2}[/tex]
[tex]\implies \dfrac{80}{64}=\dfrac{100}{b^2}[/tex]
[tex]\implies \dfrac{5}{4}=\dfrac{100}{b^2}[/tex]
[tex]\implies b^2=\dfrac{4 \cdot 100}{5}[/tex]
[tex]\implies b^2=\dfrac{400}{5}[/tex]
[tex]\implies b^2=80[/tex]
Therefore, the equation of the hyperbola is:
[tex]\boxed{\dfrac{(x-1)^2}{64}-\dfrac{(y-4)^2}{80}=1}[/tex]
To find the foci, first find c where c² = a² + b²:
[tex]\implies c=\sqrt{a^2+b^2}[/tex]
[tex]\implies c=\sqrt{64+80}[/tex]
[tex]\implies c=\sqrt{144}[/tex]
[tex]\implies c=12[/tex]
Substitute the found value of c, along with the values of h and k into the foci formula:
[tex]\implies (h+c, k)=(1+12, 4)=(13,4)[/tex]
[tex]\implies (h-c,k)=(1-12,4)=(-11,4)[/tex]
Therefore, the foci are (-11, 4) and (13, 4).